Application: Cartan’s Theorem

Let \((M,g)\) be a smooth Riemannian manifold, with Levi–Civita connection \(\nabla\). See the connection notes for the notation.

The differential of the exponential map is described by Jacobi fields, and the Jacobi equation is determined by curvature along radial geodesics. This suggests a way to compare two metrics: identify their tangent spaces at chosen base points, extend that identification by parallel transport, and compare their curvature tensors.

Cartan’s theorem makes this precise. Under the stated curvature compatibility, the map obtained by matching exponential coordinates is a local isometry. The proof compares Jacobi fields with corresponding initial data.

Theorem 1 (Cartan's local isometry theorem). Let \((M,g),(\widetilde M,\widetilde g)\) have the same dimension, and let \(i:T_pM\to T_{\widetilde p}\widetilde M\) be a linear isometry. Let \(U\subset T_pM\) be an open star-shaped neighborhood of zero such that \(\exp_p:U\to V\) is a diffeomorphism and \(\exp_{\widetilde p}\) is defined on \(i(U)\). Define \(f=\exp_{\widetilde p}\circ i\circ\exp_p^{-1}\).

For \(v\in U\), set \(\gamma_v(t)=\exp_p(tv)\), \(\widetilde\gamma_v(t)=\exp_{\widetilde p}(tiv)\), \(0\leq t\leq1\). Let \(P_t,\widetilde P_t\) be parallel transports from their initial points, and define

\[A_{v,t}=\widetilde P_t\circ i\circ P_t^{-1}:T_{\gamma_v(t)}M\longrightarrow T_{\widetilde\gamma_v(t)}\widetilde M.\]

Suppose for every such radial segment, every \(t\), and all \(x,y,u,w\in T_{\gamma_v(t)}M\),

\[\langle R(x,y)u,w\rangle=\langle\widetilde R(A_{v,t}x,A_{v,t}y)A_{v,t}u,A_{v,t}w\rangle.\tag{1}\]

Then \(f\) is a local isometry and \(df_p=i\). It suffices to impose (1) at the endpoint of every radial segment, since the truncated segments give the intermediate conditions.

Proof. Fix \(v\in U\) and abbreviate \(A_t=A_{v,t}\). It is a linear isometry by construction, and parallel transport of the initial velocity gives \(A_t\gamma_v'=\widetilde\gamma_v'\). It also commutes with covariant differentiation along these curves: in parallel frames both operations merely differentiate the same coefficient functions.

For any \(w\in T_pM\), varying the initial velocity yields the Jacobi field

\[J_w(t)=(d\exp_p)_{tv}(tw),\qquad J_w(0)=0,\quad J_w'(0)=w.\]

The curvature hypothesis and nondegeneracy of the inner product imply the vector identity

\[A_t R(\gamma_v',J_w)\gamma_v'=\widetilde R(\widetilde\gamma_v',A_tJ_w)\widetilde\gamma_v'.\]

Thus \(A_tJ_w\) satisfies the target Jacobi equation, with initial data \(0,iw\). By uniqueness it equals \(\widetilde J_w(t)=(d\exp_{\widetilde p})_{tiv}(tiw)\).

For clarity, in parallel orthonormal frames \(E_j,\widetilde E_j=A_tE_j\), write \(J_w=y^jE_j\). Both equations are precisely

\[(y^j)''+\sum_i\langle R(\gamma_v',E_i)\gamma_v',E_j\rangle y^i=0,\]

with corresponding initial coefficients. This states explicitly why matching curvature and parallel transport, rather than only curvature at the base point, determine the metrics here.

At \(t=1\), the equality of Jacobi fields gives

\[(d\exp_{\widetilde p})_{iv}\,i=A_1(d\exp_p)_v.\]

The source differential is invertible on \(U\), so the chain rule yields

\[df_{\exp_p(v)}=(d\exp_{\widetilde p})_{iv}\,i\,(d\exp_p)_v^{-1}=A_1.\]

Therefore \(df\) preserves all inner products and is invertible; the inverse function theorem gives a local diffeomorphism preserving the metric. At \(v=0\), both exponential differentials are identities, so \(df_p=i\). No nonsingularity of the target exponential was assumed—it follows from this calculation. ◻

Remark 2. The argument extends globally when both exponential maps provide global coordinates. In that case the proof of Theorem 1 shows that if \(\exp_p\) and \(\exp_{\widetilde p}\) are diffeomorphisms, then \(f\) is defined on all of \(M\), and is an isometry.

The curvature compatibility must then hold along every complete radial segment. The composition of the two global exponential diffeomorphisms and \(i\) is bijective with smooth inverse \(\exp_p\circ i^{-1}\circ\exp_{\widetilde p}^{-1}\). The already proved equality of metrics therefore makes it a global Riemannian isometry, and hence a distance isometry because lengths of all curves are preserved in both directions.

Reading: Carmo (1992), Riemannian geometry.